Dear FOM members,
I do not know whether this debate is interesting for other members,
and I would avoid a personal debate with Ketland. For that reason this is my
last personal reply, and I would be happy to know opinions about the point I
underlined in my last posting (and at the end of this): the relationship
between foundations and history of mathematics.
By the way, few apologies and remarks about Ketland's reply. First,
Ketland writes:
Mathematical objects do
not "evolve" - it is our understanding of them which evolves.
Answer: Right. I apologize for the wrong term "objects". The right term
could be "ideas". Another Ketland's remark is:
If mathematical objects do not exist objectively outside us, then
(presumably) Borzacchini's counterthesis is that mathematical objects are
"subjective" and somehow exist "inside" our heads? Is that right?
Answer: Time-out. I acknowledge I am far from being sure of anything about
these philosophical questions. My starting point was just history of
mathematics, and from this view I underline the weaknesses of the different
philosophical views.
I consider mathematical platonism a respectable view, certainly more solid
than others, and exactly for its solidity I think it deserves a serious
historical check.
Finally, there is just one Ketland's remark that it is useful to analyse:
And the actual (Platonic) continuum itself - the set of equivalence classes
of
Cauchy sequences of ratios (or some equivalent order-complete real
closed field) - doesn't exist in any objective sense? So, this is subjective
idealism again.
Answer: About this point I can be quite sharp: before Aristotle, 'continuum'
was only a soup of paradoxes concerning being/not-being, one/many, infinite;
and even the difference between 'monad' and 'point' was only in their
respective 'not-having/having position'.
In addition, Plato employs the term 'continuous' just in a naive
common language usage (meaning "without interruptions"), so that even a
sequence of consecutive integers was 'continuous'.
Hence, properly speaking, the "actual (Platonic) continuum itself" is just
absurd.
If you write 'Platonic' to mean 'xx century mathematical platonist', I
simple do not know whether "the set of equivalence classes of Cauchy
sequences of ratios (or some equivalent order-complete real closed field)
exists", but I think that such a possible 'lack of faith' does not mean
'subjective idealism'. Simply my "mind's eye" is myopic, but I think that
very few "mind's eyes" are so sharp to see such a thing somewhere, maybe
through a sort of 'scottish fog'.
I would like to understand just one point: Galileo's "mind's eye" saw the
continuum as an infinite mixture of points and gaps, then: did such a
continuum 'exist' four centuries ago? If the answer is positive, it means
that the 'continuum' exists but evolves, if the answer is negative, consider
that even tomorrow a great mathematician could give another definition of
'continuity' incompatible with ours, thus causing the sudden vanishing of
"the set of equivalence classes of Cauchy sequences of ratios"!
By the way, please read carefully Brown's characterization: doesn't it sound
a little Hermes-Trismegistus-like?
Luigi Borzacchini